Markov properties in presence of measurement noise

Recently, several powerful tools for the reconstruction of stochastic differential equations from measured data sets have been proposed [e.g., Siegert et al., Phys. Lett. A 243, 275 (1998); Hurn et al., J. Time Series Anal. 24, 45 (2003)]. Efficient application of the methods, however, generally requires Markov properties to be fulfilled. This constraint typically seems to be violated on small scales, which frequently is attributed to physical effects. On the other hand, measurement noise such as uncorrelated measurement and discretization errors has large impacts on the statistics of measurements on small scales. We demonstrate that the presence of measurement noise, likewise, spoils Markov properties of an underlying Markov process. This fact is promising for the further development of techniques for the reconstruction of stochastic processes from measured data, since limitations at small scales might stem from artificial noise sources rather than from intrinsic properties of the dynamics of the underlying process. Measurement noise, however, can be controlled much better than the intrinsic dynamics of the underlying process.

Figure 1

Example for the analysis of Markov properties by means of graphical inspection of transition PDFs, prepared by Wächter et al. [10]. Test for Markov properties of Au film data for two different scale separations $\Delta r=14\mathrm{nm}$ (left-hand side) and $35\mathrm{nm}$ (right-hand side), where $\Delta r=r_3-r_2=r_2-r_1$. In both cases $r_2=169\mathrm{nm}$. In each case a contour plot of conditional probabilities $P(h_1,r_1\mid h_2,r_2)$ (dashed lines) and $P(h_1,r_1\mid h_2,r_2;h_3=0,r_3)$ (solid lines) is shown in the top panel. Contour levels differ by a factor of 10, with an additional level at $p=0.3$. Below the top panels in each case, two one-dimensional cuts at $h_2\approx \pm {\sigma }_{\infty }$ are shown with $P(h_1,r_1\mid h_2,r_2)$ as dashed lines and $P(h_1,r_1\mid h_2,r_2;h_3=0,r_3)$ as circles. From the deviations of the PDFs for $\Delta r=14\mathrm{nm}$ (left-hand side) it becomes evident, that Markov properties are not fulfilled in this case. They might, however, be valid for $\Delta r=35\mathrm{nm}$ (right-hand side).

Figure 2

Example of a process $x\left(t\right)$ according to Eq. (8), that is affected by strong discretization noise. Here, the faint line specifies the original process $x\left(t\right)$, whereas the bold line depicts the evolution $y\left(t\right)$, that eventually is obtained from the measurement due to discretization errors.

Figure 3

Detail of a sample path of the stochastic process (17) for the parameters $(\gamma ,D)=(0.75,0.1)$. The occurrence of distinct peaks is characteristic for multiplicative stochastic processes.

Figure 4

Test for Markov properties for simulated samples A without measurement noise (LHS) and B with artificial Gaussian measurement noise with variance $2.25\times {10}^{-2}$ (RHS), respectively. In the upper panels, the conditional transition PDFs for $\tau =0.1$ (solid contour lines) are compared with the ones obtained for the same time increment by numerical integration of the Chapman-Kolmogorov equation (2) for transition PDFs for increment $\tau ∕2$ (dashed contour lines). Contour lines are placed at the levels 10, 1, 0.1, and 0.01. In the lower panels, a cross section of the transition PDF at $x\left(t\right)=1$ is depicted. For reasons of clearness, circles have been added to the dashed lines corresponding to the data set B. Perfect coincidence of the PDFs is observed for A, whereas in case of B systematic deviations become evident. Consequently, Markov properties are spoiled by the artificial measurement noise of sample B.